Fifty students, each with .75, visited the arcade of Example 3.27.

Question

Seventeen of the students played each of the three computer games, and 37 of them played at least two of them. No student played any other game at the arcade, nor did any student play a given game more than once. Each game costs .25 to play, and the total proceeds from the student visit were $24.25. How many of these students preferred to watch and played none of the games?

Answer 1

Denote the three games as X, Y, Z.

Consider 8 regions as follows in terms of the number of players:

A  Players of X only

B Players of Y only

C Players of Z only

D Players of X and Y

E Players of X and Z

F Players of Y and Z

G Players of all three games

H Observers only (to be found)

From the given facts:

G=17

A+B+C+D+E+F+G+H=50

A+D+E+G played X

B+D+F+G played Y

C+E+F+G played Z

50-37=13 were observers or played only one game: A+B+C+H=13

24.25/0.25=97 games played: (A+D+E+G)+(B+D+F+G)+(C+E+F+G)=97,

A+B+C+2(D+E+F)+3G=97,

13-H+2(D+E+F)+51=97,

2(D+E+F)-H=33, D+E+F=(H+33)/2

A+B+C+D+E+F+G+H=50=13-H+(H+33)/2+17+H,

(H+33)/2=20,

H+33=40, H=7.

So 7 students watched and didn't play.